Contents

Lectures on Stochastic Analysis

Thomas G. Kurtz Departments of Mathematics and Statistics

University of Wisconsin - Madison Madison, WI 53706-1388

Revised September 7, 2001 Minor corrections August 23, 2007

Contents

1 Introduction.

4

2 Review of probability.

5

2.1 Properties of expectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Convergence of random variables. . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Convergence in probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Information and independence. . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Conditional expectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Continuous time stochastic processes.

13

3.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Filtrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Stopping times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Martingales.

18

4.1 Optional sampling theorem and Doob's inequalities. . . . . . . . . . . . . . . 18

4.2 Local martingales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Quadratic variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4 Martingale convergence theorem. . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Stochastic integrals.

22

5.1 Definition of the stochastic integral. . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Conditions for existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3 Semimartingales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.4 Change of time variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.5 Change of integrator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.7 Approximation of stochastic integrals. . . . . . . . . . . . . . . . . . . . . . . 33

5.8 Connection to Protter's text. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1

6 Covariation and It^o's formula.

35

6.1 Quadratic covariation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.2 Continuity of the quadratic variation. . . . . . . . . . . . . . . . . . . . . . . 36

6.3 Ito's formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.4 The product rule and integration by parts. . . . . . . . . . . . . . . . . . . . 40

6.5 It^o's formula for vector-valued semimartingales. . . . . . . . . . . . . . . . . 41

7 Stochastic Differential Equations

42

7.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7.2 Gronwall's inequality and uniqueness for ODEs. . . . . . . . . . . . . . . . . 42

7.3 Uniqueness of solutions of SDEs. . . . . . . . . . . . . . . . . . . . . . . . . 44

7.4 A Gronwall inequality for SDEs . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.5 Existence of solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.6 Moment estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8 Stochastic differential equations for diffusion processes.

53

8.1 Generator for a diffusion process. . . . . . . . . . . . . . . . . . . . . . . . . 53

8.2 Exit distributions in one dimension. . . . . . . . . . . . . . . . . . . . . . . . 54

8.3 Dirichlet problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.4 Harmonic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.5 Parabolic equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.6 Properties of X(t, x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.7 Markov property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.8 Strong Markov property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8.9 Equations for probability distributions. . . . . . . . . . . . . . . . . . . . . . 59

8.10 Stationary distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.11 Diffusion with a boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

9 Poisson random measures

63

9.1 Poisson random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

9.2 Poisson sums of Bernoulli random variables . . . . . . . . . . . . . . . . . . . 64

9.3 Poisson random measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

9.4 Integration w.r.t. a Poisson random measure . . . . . . . . . . . . . . . . . . 66

9.5 Extension of the integral w.r.t. a Poisson random measure . . . . . . . . . . 68

9.6 Centered Poisson random measure . . . . . . . . . . . . . . . . . . . . . . . . 71

9.7 Time dependent Poisson random measures . . . . . . . . . . . . . . . . . . . 74

9.8 Stochastic integrals for time-dependent Poisson random measures . . . . . . 75

10 Limit theorems.

79

10.1 Martingale CLT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

10.2 Sequences of stochastic differential equations. . . . . . . . . . . . . . . . . . 82

10.3 Approximation of empirical CDF. . . . . . . . . . . . . . . . . . . . . . . . . 83

10.4 Diffusion approximations for Markov chains. . . . . . . . . . . . . . . . . . . 83

10.5 Convergence of stochastic integrals. . . . . . . . . . . . . . . . . . . . . . . . 86

2

11 Reflecting diffusion processes.

87

11.1 The M/M/1 Queueing Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 87

11.2 The G/G/1 queueing model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

11.3 Multidimensional Skorohod problem. . . . . . . . . . . . . . . . . . . . . . . 89

11.4 The Tandem Queue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

12 Change of Measure

93

12.1 Applications of change-of-measure. . . . . . . . . . . . . . . . . . . . . . . . 93

12.2 Bayes Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

12.3 Local absolute continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

12.4 Martingales and change of measure. . . . . . . . . . . . . . . . . . . . . . . . 95

12.5 Change of measure for Brownian motion. . . . . . . . . . . . . . . . . . . . . 96

12.6 Change of measure for Poisson processes. . . . . . . . . . . . . . . . . . . . . 97

13 Finance.

99

13.1 Assets that can be traded at intermediate times. . . . . . . . . . . . . . . . . 100

13.2 First fundamental "theorem". . . . . . . . . . . . . . . . . . . . . . . . . . . 103

13.3 Second fundamental "theorem". . . . . . . . . . . . . . . . . . . . . . . . . . 105

14 Filtering.

106

15 Problems.

109

3

1 Introduction.

The first draft of these notes was prepared by the students in Math 735 at the University of Wisconsin - Madison during the fall semester of 1992. The students faithfully transcribed many of the errors made by the lecturer. While the notes have been edited and many errors removed, particularly due to a careful reading by Geoffrey Pritchard, many errors undoubtedly remain. Read with care.

These notes do not eliminate the need for a good book. The intention has been to state the theorems correctly with all hypotheses, but no attempt has been made to include detailed proofs. Parts of proofs or outlines of proofs have been included when they seemed to illuminate the material or at the whim of the lecturer.

4

2 Review of probability.

A probability space is a triple (, F, P ) where is the set of "outcomes", F is a -algebra of "events", that is, subsets of , and P : F [0, ) is a measure that assigns "probabilities" to events. A (real-valued) random variable X is a real-valued function defined on such that for every Borel set B B(R), we have X-1(B) = { : X() B} F . (Note that the Borel -algebra B(R)) is the smallest -algebra containing the open sets.) We will occasionally also consider S-valued random variables where S is a separable metric space (e.g., Rd). The definition is the same with B(S) replacing B(R).

The probability distribution on S determined by

?X(B) = P (X-1(B)) = P {X B}

is called the distrbution of X. A random variable X has a discrete distribution if its range is countable, that is, there exists a sequence {xi} such that P {X = xi} = 1. The expectation of a random variable with a discrete distribution is given by

E[X] = xiP {X = xi}

provided the sum is absolutely convergent. If X does not have a discrete distribution, then

it can be approximated

by random variables with discrete distributions.

Define Xn =

k+1 n

and

Xn

=

k n

when

k n

<

X

k+1 n

,

and

note

that

Xn

<

X

Xn

and

|X n

- Xn|

1 n

.

Then

E [X ]

lim

n

E [X n ]

=

lim

E [X n ]

provided E[Xn] exists for some (and hence all) n. If E[X] exists, then we say that X is integrable.

2.1 Properties of expectation.

a) Linearity: E[aX + bY ] = aE[X] + bE[Y ] b) Monotonicity: if X Y a.s then E[X] E[Y ]

2.2 Convergence of random variables.

a) Xn X a.s. iff P { : limn Xn() = X()} = 1.

b) Xn X in probability iff > 0, limn P {|Xn - X| > } = 0.

c) Xn converges to X in distribution (denoted Xn X) iff limn P {Xn x} = P {X x} FX(x) for all x at which FX is continuous.

Theorem 2.1 a) implies b) implies c). Proof. (b c) Let > 0. Then

P {Xn x} - P {X x + } = P {Xn x, X > x + } - P {X x + , Xn > x} P {|Xn - X| > }

and hence lim sup P {Xn x} P {X x + }. Similarly, lim inf P {Xn x} P {X x - }. Since is arbitrary, the implication follows.

5

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